P l a n e G e o m e t r y
An Adventure in Language and Logic
RELATIONS AMONG THE
|If one side of a triangle is extended, then the exterior angle is greater than either of the opposite interior angles.|
|Let ABC be a triangle, and let one side of it BC be extended to D;
then the exterior angle ACD is greater than either of the opposite interior angles, ABC, CAB.
|(We now draw triangle BFC in such a way that triangles BAE, ECF will be congruent; that makes angle BAE equal to angle ECF. Eventually, Proposition 27, that will imply that the lines BA, CF are parallel.)|
|Bisect AC at E;||(I. 10)|
|draw BE and extend it in a straight line to F, making EF equal
|draw FC, and extend AC to a point G.|
|Then, since AE is equal to EC, and BE to EF,|
|the two sides AE, EB are equal to the two sides CE, EF respectively;|
|and angle AEB is equal to angle CEF,|
|because they are vertical angles.||(I. 15)|
|Therefore triangles AEB, CEF are congruent,||(S.A.S.)|
|and the remaining angles are equal to the remaining angles, namely those that are opposite the equal sides:|
|angle BAE is equal to angle ECF.|
|But angle ECD is greater than angle ECF;||(Axiom 5)|
|therefore angle ACD is greater than angle BAE.|
|If we bisect BC, in the same way then we can prove that angle BCG,|
|that is, angle ACD,||(I. 15)|
|is greater than angle ABC as well.|
|Therefore, if one side of a triangle etc. Q.E.D.|
|Any two angles of a triangle are together less than two right angles.|
|Let ABC be a triangle;
then any two of its angles are together less than two right angles.
|Extend BC to D.|
|Then, since ACD is the exterior angle of the triangle,
it is greater than opposite interior angle ABC.
|To each of those join angle ACB.|
|angles ACD, ACB are greater than angles ABC, ACB.|
|But angles ACD, ACB are together equal to two right angles.||(I. 13)|
|two right angles are greater than angles ABC, ACB.|
|Therefore angles ABC, ACB are together less than two right angles.|
|In the same way we could show that angles BAC, ACB, and also angles CAB, ABC, are together less than two right angles.|
|Therefore, any two angles etc. Q.E.D.|
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